Tresca Friction

As a preliminary step before going into Coulomb friction, we focus on the numerical approximation of Tresca friction in this chapter. The Signorini problem with Tresca friction leads to a well-posed variational inequality of the second kind. As a result, this setting makes easier the mathematical analysis of some numerical approximations, as it was the case for the (frictionless) Signorini problem. However, as we will see, new difficulties appear in the derivation of a priori error estimates, and, for some classes of methods, it is still an open problem to derive estimates with the optimal rates. We will show that, still today, optimal convergence rates can be obtained solely for Nitsche and penalty methods. We will try to explain at least the sources of difficulties for some other techniques. Moreover, in the next chapter, we will see that some existence results for discrete Coulomb problems can be established with a Brouwer fixed point argument. For this latter, we define a function that involves a Tresca problem. As a consequence, the results of discrete well-posedness presented here will be helpful for this purpose.